How do you add a vertex to a shape?
Enter Edit Mode by pressing Tab with the object selected. Then, press 1 to enter vertex selection mode. Select any vertex, then press E to extrude. This will add another vertex that you can play with.

Since you'll probably want to connect that vertex back to the mesh somehow, you can fill in space between two vertices by first selecting them both (hold Shift while selecting to do that), then pressing F. This will add an edge between them.

If you want to fill in a face (in the case that you have a wireframe but no face), press 2 to enter Edge Select mode, select the surrounding edges of the shape, and press F.